Gears is pretty easy. Takes a special tool or two but it isn't rocket surgery. The cassette is a wear part, they're meant to be replaced. Which tools you need depends on which hub you've got. Any bike shop can get you sorted after you convince them that you really will not spend $400 to buy a new bike that will do exactly what this one already does and will do better with a cassette and tire change.
Nashbar can sell you a new 6s cassette for like $15 that will even shift better. If you come upon a worthy 7s wheel it'll slip right in, flip the lever on your shifter to friction mode and don't look back.
The tools aren't expensive. If you're going to get into this to any extent you might want to pick up one of those semi-crappy $40 tool kits that'll do ya until you see what tools you need the good version of.
A new chain goes with a new cassette. You'll want to shorten the old one anyway to install a smaller set of cogs. You can get the cheap KMC or Sram chain and be good.
I see no reason to replace something that works. You've got a decent 80s MTB which means it's a decent all around bike for almost any purpose other than racing. Or fast road riding.
So what's the cute 4x10 brown amp costarring with the Trek?